Understanding Engineering Mathematics

8.3.2 Differentiation from first principles

(i) 3 (ii) 2x+ 2 (iii) 3x^2 (iv) −sinx

8.3.3 Standard derivatives

A.(i) ex (ii) −sinx (iii) 31x^30 (iv)

1

x

(v) cosx

(vi)

1

3

x−^2 /^3 (vii) sec^2 x (viii) −

3

x^4

B.(i)

x^5

5

+C (ii) sinx+C (iii) ex+C (iv) −cosx+C

(v) C−

1

3 x^3

(vi)

2

3

x^3 /^2 +C (vii) lnx+C (viii) C

(ix) tanx+C

8.3.4 Rules of differentiation

A.(i) secxtanx (ii) −cosecxcotx (iii) −cosec^2 x

(iv) sinhx (v) coshx (vi) sech^2 x

(vii) −cosechxcothx (viii) −sechxtanhx (ix) −cosech^2 x

B.(i) tanx (ii) cotx (iii) secx

(iv) −cosecx (v) tanhx (vi) cothx

C.(i) 7x^6 − 10 x^4 + 4 x^3 − 2 x (ii) 2xtanx+(x^2 + 2 )sec^2 x

(iii)

x^2 + 1 − 2 x^2 lnx

x(x^2 + 1 )^2

(iv) ( 3 x^2 − 2 )ex

(^3) − 2 x
(v) −
1
(x^2 − 1 )^3 /^2
(vi)
−sinx
cosx+ 1
(vii) −
1
x^2
cos
(
x+ 1
x
)
(viii) secxtan^2 x+sec^3 x (ix) 6e^6 x (x) ex(x+ 1 )
(xi) − 2 xe−x
2
(xii)
1
x
(xiii) ex(lnx+
1
x
)
(xiv) 2
8.3.5 Implicit differentiation
A.(i) −
1
√
1 −x^2
(ii)
1
1 +x^2
(iii) πxlnπ
B.(i) 0 (ii)
5
2
C. −
4
9